Prove that $\delta_{x_0}\in W^{-m,p}(\Omega)$

Question

Prove that $\delta_{x_0}\in W^{-m,p}(\Omega)$

here $\delta_{x_0} (\phi)=\phi(x_0) $ how to prove this

and this is true only for large negative order for $m>\frac{N}{p}$

Answer 1

$$|\delta_{x_0}(\phi)| = |\phi(x_0)| \leq \|\phi\|_{C(\Omega)} \leq C \|\phi\|_{W^{m,p}}$$

where we used $W^{m,p}(\Omega) \hookrightarrow C(\Omega)$ for $mp>N$. (assuming a Lipschitz domain)